DOCSLIB.ORG

Comparisons of Relative BV-Capacities and Sobolev Capacity in Metric Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Comparisons of Relative BV-Capacities and Sobolev Capacity in Metric Spaces Comparisons of relative BV-capacities and Sobolev capacity in metric spaces H. Hakkarainen∗ Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland. N. Shanmugalingam Department of Mathematical Sciences, P.O.Box 210025, University of Cincinnati, Cincinnati, OH 45221{0025, U.S.A. Abstract We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and support- ing a weak (1; 1)-Poincar´einequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for 1 < p < 1 to also cover the more geometric case p = 1. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity. Keywords: Capacity, Functions of Bounded Variation, Analysis on Metric Measure Spaces 2010 MSC: 28A12, 26A45, 30L99 1. Introduction In this article we study connections between different 1-capacities and BV-capacities in the setting of metric measure spaces. We obtain characterizations for the 1-capacity of a condenser, extending results known for p > 1 to also cover the more geometric case p = 1. Furthermore, we study how the different versions of BV-capacity, corresponding to various pointwise requirements that the capacity test functions need to fulfill, relate to each other. One difficulty that occurs when working with minimization problems on the space W 1;1(Rn) is the lack of reflexivity. Indeed, the methods used to develop the theory of p-capacity are closely related to those used in certain variational minimization problems; in such problems reflexivity or the weak compactness property of the function space W 1;p(Rn) when p > 1 usually plays an important role. One possible way to deal with ∗Corresponding author Email addresses: [email protected] (H. Hakkarainen), [email protected] (N. Shanmugalingam) Preprint submitted to Elsevier 28th March 2011 this lack of reflexivity is to consider the space BV(Rn), that is, the class of functions of bounded variation. This wider class of functions provides tools, such as lower semiconti- nuity of the total variation measure, that can be used to overcome the problems caused by the lack of reflexivity in the arguments. This approach was originally used to study variational 1-capacity in the Euclidean case in [1], [2], and [3]. The article [4] showed that similar approach could also be used to study variational 1-capacity in the setting of metric measure spaces. In [4] the main tool in obtaining a connection between the 1-capacity and BV-capacity was the metric space version of Gustin's boxing inequality, see [5]. Since then, this strategy has been used in [6] to study a version of BV-capacity and Sobolev 1-capacity in the setting of metric measure spaces. Since the case p = 1 corresponds to geometric objects in the metric measure space such as sets of finite perimeter and minimal surfaces, it behooves us to understand this case. A study of potential theory in the case p = 1 for the setting of metric measure spaces was begun in the papers [6], [4], and [7], and we continue the study in this note. In the theory of calculus of variations, partial differential equations and potential theory many interesting features of Sobolev functions are measured in terms of capacity. Just as sets of measure zero are associated with the Lp-theory, sets of p-capacity zero should be understood in order to deal with the Sobolev space W 1;p(Rn). For 1 < p < 1 the p-capacity of sets is quite well-understood; see for example [8], [9], [10] for the Euclidean setting, and in the more general metric measure space setting, [11], [12], [13], [14], [15], [16], [17], and the references therein. The situation corresponding to the case p = 1 is not so well-understood. In the setting of p = 1 there are two natural function spaces to consider: Sobolev type spaces, and spaces of functions of bounded variation. Since these two spaces are interrelated, it is natural to expect that the corresponding capacities are related as well. However, these two function spaces are fundamentally different in nature. Functions in the Sobolev class W 1;1(Rn) have quasicontinuous representatives, whereas functions of bounded variation need not have quasicontinuous representatives, and in fact sometimes exhibit jumps across sets of nonzero codimension one Hausdorff measure; see for exam- ple [18]. Because of the quasicontinuity of Sobolev type functions, one can either insist on the test functions to have value one in a neighborhood of the set whose capacity is being computed, or merely require the test functions to have value 1 on the set. Such flexibility is not available in computing the BV-capacities and hence different point-wise requirements on the test functions might lead to different types of BV-capacity. Thus, corresponding to the BV-class there is more than one possible notion of capacity and it is nontrivial to even know which versions of BV-capacities are equivalent. We point out here that the analog of Sobolev spaces considered in this paper, called the Newton-Sobolev spaces, consist automatically only of quasicontinuous functions when the measure on X is doubling and the space supports a (1; 1)-Poincar´einequality. The primary goal of this note is to compare these different notions of capacity related to the BV-class, and to the capacity related to the Newton-Sobolev space N 1;1(X). In Section 2 we describe the objects considered in this paper, and then in Section 3 we show that the 1-modulus of the family of curves in a domain Ω in the metric space, connecting two nonempty pairwise disjoint compact sets E; F ⊂ Ω, is the same as the Newton{Sobolev 1-capacity and local Lipschitz capacity of the condenser (E; F; Ω). Our arguments are based on constructing appropriate test functions. In Section 4 we consider 2 three alternative notions of variational BV-capacity of a compact set K ⊂ Ω, and show that these notions are comparable to each other and to the variational Sobolev 1-capacity of K relative to Ω. We combine tools such as discrete convolution and boxing inequality to obtain the desired results. In Section 5 we consider total capacities where we minimize the norm of the test functions rather than their energy seminorm. We consider these quantities of more general bounded sets - related to both BV-functions and to Sobolev functions - and compare them to their variational capacities. In the classical Euclidean setting some of these results are known, for example from [2], but even in the weighted Euclidean setting and the Carnot groups the results of this paper are new. 2. Preliminaries In this article X = (X; d; µ) is a complete metric measure space. We assume that µ is a Borel regular outer measure that is doubling, i.e. there is a constant CD, called the doubling constant of µ, such that µ(2B) ≤ CDµ(B) for all balls B of X. Furthermore, it is assumed that 0 < µ(B) < 1 for all balls B ⊂ X. These assumptions imply that the metric space X is proper, that is, closed and bounded sets are compact. In this article, by a path we mean a rectifiable nonconstant continuous mapping from a compact interval to X. The standard tool that is used to measure path families is the following. Definition 2.1. Let Ω ⊂ X be an open connected set. The p-modulus of a collection of paths Γ in Ω for 1 ≤ p < 1 is defined as Z p Modp(Γ) = inf % dµ % Ω where the infimum is taken over all nonnegative Borel-measurable functions % such that R γ % ds ≥ 1 for each γ 2 Γ. We use the definition of Sobolev spaces on metric measure space X based on the notion of p-weak upper gradients, see [11], [19]. Note that instead of the whole space X, we can consider its open subsets in the definitions below. Definition 2.2. A nonnegative Borel function g on X is an upper gradient of an extended real valued function u on X if for all paths γ, Z ju(x) − u(y)j ≤ g ds; (1) γ R whenever both u(x) and u(y) are finite, and γ g ds = 1 otherwise. Here x and y denote end points of γ. Let 1 ≤ p < 1. If g is a nonnegative measurable function on X, and if the integral in (1) is well defined and the inequality holds for p-almost every path, then g is a p-weak upper gradient of u. 3 The phrase that inequality (1) holds for p-almost every path with 1 ≤ p < 1 means that it fails only for a path family with zero p-modulus. Many usual rules of calculus are valid for upper gradients as well. For a good overall reference interested readers may see [20], [21], and [22]. p If u has a p-weak upper gradient g 2 Lloc(X), then there is a minimal p-weak upper gradient gu such that gu ≤ g µ-almost everywhere for every p-weak upper gradient g of u, see [21] and [22]. When Ω is an open subset of Euclidean space equipped with the Euclidean metric and Lebesgue measure, the Sobolev type space considered below coincides with the space of quasicontinuous representatives of classical Sobolev functions. In the setting of weighted Euclidean spaces with p-admissible weights, or the Carnot-Carath´eodory spaces, the corresponding Sobolev spaces coincide in an analogous manner with the Sobolev type space considered below.
Load more

Recommended publications
  • Theory of Capacities Annales De L’Institut Fourier, Tome 5 (1954), P
    ANNALES DE L’INSTITUT FOURIER GUSTAVE CHOQUET Theory of capacities Annales de l’institut Fourier, tome 5 (1954), p. 131-295 <http://www.numdam.org/item?id=AIF_1954__5__131_0> © Annales de l’institut Fourier, 1954, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THEORY OF CAPACITIES (l) by Gustave CHOQUETOQ. INTRODUCTION This work originated from the following problem, whose significance had been emphasized by M. Brelot and H. Cartan : Is the interior Newtonian capacity of an arbitrary borelian subset X of the space R3 equal to the exterior Newtonian capacity of X ? For the solution of this problem, I first systematically studied the non-additive set-functions, and tried to extract from their totality certain particularly interesting classes, with a view to establishing for these a theory analogous to the classical theory of measurability. I succeeded later in showing that the classical Newtonian capacity f belongs to one of these classes, more precisely: if A and B are arbitrary compact subsets of R3, then AAU^+AAflB^/^+AB). It followed from this that every borelian, a^rid even every analytic set is capacitable with respect to the Newtonian capa- city, a result which can, moreover, be extended to the capa- (') This research was supported by the United States Air Force, throught the Office of Scientific Research of the Air Research and Development Command.
    [Show full text]
  • The Dimension of Chaotic Attractors
    Physica 7D (1983) 153-180 North-Holland Publishing Company THE DIMENSION OF CHAOTIC ATTRACTORS J. Doyne FARMER Center for Nonlinear Studies and Theoretical Division, MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Edward OTT Laboratory of Plasma and Fusion Energy Studies, University of Maryland, College Park, Maryland, USA and James A. YORKE Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland, USA Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the "dimension of the natural measure", and all of the metric dimensions take on a common value, which we call the "fractal dimension". Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension. Table of contents 1.
    [Show full text]
  • Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory
    ENDS, FUNDAMENTAL TONES, AND CAPACITIES OF MINIMAL SUBMANIFOLDS VIA EXTRINSIC COMPARISON THEORY VICENT GIMENO AND S. MARKVORSEN ABSTRACT. We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotation- ally symmetric model manifold. Using the asymptotic behavior of the volumes and ca- pacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question. 1. INTRODUCTION Let M be a complete non-compact Riemannian manifold. Let K ⊂ M be a compact set with non-empty interior and smooth boundary. We denote by EK (M) the number of connected components E1; ··· ;EEK (M) of M n K with non-compact closure. Then M EK (M) has EK (M) ends fEigi=1 with respect to K (see e.g. [GSC09]), and the global number of ends E(M) is given by (1.1) E(M) = sup EK (M) ; K⊂M where K ranges on the compact sets of M with non-empty interior and smooth boundary. The number of ends of a manifold can be bounded by geometric restrictions. For ex- ample, in the particular setting of an m−dimensional minimal submanifold P which is properly immersed into Euclidean space Rn, the number of ends E(P ) is known to be re- lated to the extrinsic properties of the immersion.
    [Show full text]
  • Introduction to Heat Potential Theory
    Mathematical Surveys and Monographs Volume 182 Introduction to Heat Potential Theory Neil A. Watson American Mathematical Society http://dx.doi.org/10.1090/surv/182 Mathematical Surveys and Monographs Volume 182 Introduction to Heat Potential Theory Neil A. Watson American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 31-02, 31B05, 31B20, 31B25, 31C05, 31C15, 35-02, 35K05, 31B15. For additional information and updates on this book, visit www.ams.org/bookpages/surv-182 Library of Congress Cataloging-in-Publication Data Watson, N. A., 1948– Introduction to heat potential theory / Neil A. Watson. p. cm. – (Mathematical surveys and monographs ; v. 182) Includes bibliographical references and index. ISBN 978-0-8218-4998-9 (alk. paper) 1. Potential theory (Mathematics) I. Title. QA404.7.W38 2012 515.96–dc23 2012004904 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society.
    [Show full text]
  • On Essential Self-Adjointness, Confining Potentials & the Lp-Hardy
    Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. On Essential Self-adjointness, Confining Potentials & the Lp-Hardy Inequality A Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand A.D.Ward - New Zealand Institute of Advanced Study August 8, 2014 Abstract Let Ω be a domain in Rm with non-empty boundary and let H = −∆ + V be a 1 Schr¨odingeroperator defined on C0 (Ω) where V 2 L1;loc(Ω). We seek the minimal criteria on the potential V that ensures that H is essentially self-adjoint, i.e. that en- sures the closed operator H¯ is self-adjoint. Overcoming various technical problems, we extend the results of Nenciu & Nenciu in [1] to more general types of domain, specifically unbounded domains and domains whose boundaries are fractal. As a special case of an abstract condition we show that H is essentially self-adjoint provided that sufficiently close to the boundary 1 1 1 V (x) ≥ 1 − µ (Ω) − − − · · · ; (1) d(x)2 2 ln( d(x)−1) ln( d(x)−1) ln ln( d(x)−1) where d(x) = dist(x; @Ω) and the right hand side of the above inequality contains a finite number of logarithmic terms. The constant µ2(Ω) appearing in (1) is the variational constant associated with the L2-Hardy inequality and is non-zero if and only if Ω admits the aforementioned inequality.
    [Show full text]
  • Rectifiability Via Curvature and Regularity in Anisotropic Problems
    ©Copyright 2021 Max Goering Rectifiability via curvature and regularity in anisotropic problems Max Goering A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2021 Reading Committee: Tatiana Toro, Chair Steffen Rohde Stefan Steinerberger Program Authorized to Offer Degree: Mathematics University of Washington Abstract Rectifiability via curvature and regularity in anisotropic problems Max Goering Chair of the Supervisory Committee: Craig McKibben and Sarah Merner Professor Tatiana Toro Mathematics Understanding the geometry of rectifiable sets and measures has led to a fascinating interplay of geometry, harmonic analysis, and PDEs. Since Jones' work on the Analysts' Traveling Salesman Problem, tools to quantify the flatness of sets and measures have played a large part in this development. In 1995, Melnikov discovered and algebraic identity relating the Menger curvature to the Cauchy transform in the plane allowing for a substantially streamlined story in R2. It was not until the work of Lerman and Whitehouse in 2009 that any real progress had been made to generalize these discrete curvatures in order to study higher-dimensional uniformly rectifiable sets and measures. Since 2015, Meurer and Kolasinski began developing the framework necessary to use dis- crete curvatures to study sets that are countably rectifiable. In Chapter 2 we bring this part of the story of discrete curvatures and rectifiability to its natural conclusion by producing mul- tiple classifications of countably rectifiable measures in arbitrary dimension and codimension in terms of discrete measures. Chapter 3 proceeds to study higher-order rectifiability, and in Chapter 4 we produce examples of 1-dimensional sets in R2 that demonstrate the necessity of using the so-called \pointwise" discrete curvatures to study countable rectifiability.
    [Show full text]
  • Upper Large Deviations for the Maximal Flow Through a Domain of Bold0mu Mumu Rdrdrdrdrdrd in First Passage Percolation
    The Annals of Applied Probability 2011, Vol. 21, No. 6, 2075–2108 DOI: 10.1214/10-AAP732 c Institute of Mathematical Statistics, 2011 UPPER LARGE DEVIATIONS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF RD IN FIRST PASSAGE PERCOLATION By Raphael¨ Cerf and Marie Theret´ Universit´eParis Sud and Ecole´ Normale Sup´erieure We consider the standard first passage percolation model in the rescaled graph Zd/n for d ≥ 2 and a domain Ω of boundary Γ in Rd. Let Γ1 and Γ2 be two disjoint open subsets of Γ representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behavior of the flow φn through a discrete 1 2 version Ωn of Ω between the corresponding discrete sets Γn and Γn. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations d−1 of φn/n above a certain constant are of volume order, that is, decays exponentially fast with nd. This article is part of a larger project in which the authors prove that this constant is the a.s. limit d−1 of φn/n . 1. First definitions and main result. We use notation introduced in [8] Zd Ed and [9]. Let d ≥ 2. We consider the graph ( n, n) having for vertices Zd Zd Ed n = /n and for edges n, the set of pairs of nearest neighbors for the 1 Ed standard L norm. With each edge e in n we associate a random vari- R+ Ed able t(e) with values in .
    [Show full text]
  • The Variational 1-Capacity and BV Functions with Zero Boundary Values on Metric Spaces
    The variational 1-capacity and BV functions with zero boundary values on metric spaces ∗ Panu Lahti June 9, 2021 Abstract In the setting of a metric space that is equipped with a doubling measure and supports a Poincar´einequality, we define and study a class of BV functions with zero boundary values. In particular, we show that the class is the closure of compactly supported BV functions in the BV norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and BV analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets. 1 Introduction Spaces of Sobolev functions with zero boundary values are essential in spec- ifying boundary values in various Dirichlet problems. This is true also in arXiv:1708.09318v1 [math.MG] 30 Aug 2017 the setting of a metric measure space (X,d,µ), where µ is a doubling Radon measure and the space supports a Poincar´einequality; see Section 2 for def- initions and notation. In this setting, given an open set Ω ⊂ X, the space of Newton-Sobolev functions with zero boundary values is defined for 1 ≤ p< ∞ by 1,p 1,p N0 (Ω) := {u|Ω : u ∈ N (X) with u =0 on X \ Ω}. ∗2010 Mathematics Subject Classification: 30L99, 31E05, 26B30. Keywords : metric measure space, bounded variation, zero boundary values, variational capacity, outer capacity, quasi-semicontinuity 1 Dirichlet problems for minimizers of the p-energy, and Newton-Sobolev func- tions with zero boundary values have been studied in the metric setting in [6, 10, 11, 41].
    [Show full text]
  • Capacity Theory on Algebraic Curves and Canonical Heights Groupe De Travail D’Analyse Ultramétrique, Tome 12, No 2 (1984-1985), Exp
    Groupe de travail d’analyse ultramétrique ROBERT RUMELY Capacity theory on algebraic curves and canonical heights Groupe de travail d’analyse ultramétrique, tome 12, no 2 (1984-1985), exp. no 22, p. 1-17 <http://www.numdam.org/item?id=GAU_1984-1985__12_2_A3_0> © Groupe de travail d’analyse ultramétrique (Secrétariat mathématique, Paris), 1984-1985, tous droits réservés. L’accès aux archives de la collection « Groupe de travail d’analyse ultramétrique » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Groupe d’ étude d ~ Analyse ultrametrique 22-01 (Y. AMICE, G. CHRISTOL, P. ROBBA) 12e année, 1984/85, n° 22, 17 p. 13 mai 1985 CAPACITY THEORY ON ALGEBRAIC CURVES AND CANONICAL HEIGHTS by Robert RUMELY This note outlines a theory of capacity for adelic sets on algebraic curves. It was motivated by a paper of D. CANTOR where the theory was developed for P~ . Complete proofs of all assertions are given in a manuscript [R], which I hope to publish in the Springer-Verlag Lecture Notes in Mathematics series. The capacity is a measure of the size of a set which is defined geometrically but has arithmetic consequences. (It goes under several names in the literature, including "Transfinite diameter", "Tchebychev constant",, and "Robbins constant",, depending on the context.) The introduction to Cantor’s paper contains several nice applications, which I encourage the reader to see.
    [Show full text]
  • On the Dual Formulation of Obstacle Problems for the Total Variation and the Area Functional
    On the dual formulation of obstacle problems for the total variation and the area functional Christoph Scheven∗ Thomas Schmidty August 9, 2017 Abstract We investigate the Dirichlet minimization problem for the total variation and the area functional with a one-sided obstacle. Relying on techniques of convex analysis, we identify certain dual maximization problems for bounded divergence- measure fields, and we establish duality formulas and pointwise relations between (generalized) BV minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of BV minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as BV obstacles, thin obstacles, and boundary obstacles, and they include information on exceptional sets and up to the boundary. As a side benefit, in some cases we also obtain assertions on the limit behavior of p-Laplace type obstacle problems for p & 1. On the technical side, the statements and proofs of our results crucially de- pend on new versions of Anzellotti type pairings which involve general divergence- measure fields and specific representatives of BV functions. In addition, in the proofs we employ several fine results on (BV) capacities and one-sided approxi- mation. Contents 1 Introduction2 2 Preliminaries7 2.1 General notation...............................7 2.2 Domains....................................8 2.3 Representatives of BV functions......................8 2.4 Admissible function classes.........................8 2.5 Anzellotti type pairings for divergence-measure fields...........9 2.6 De Giorgi's measure............................. 11 2.7 Total variation and area functionals for relaxed obstacle problems.... 13 2.8 Capacities and quasi (semi)continuity..................
    [Show full text]
  • On Hilbert Boundary Value Problem for Beltrami Equation
    Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 45, 2020, 957–973 ON HILBERT BOUNDARY VALUE PROBLEM FOR BELTRAMI EQUATION Vladimir Gutlyanskii, Vladimir Ryazanov, Eduard Yakubov and Artyem Yefimushkin National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics Generala Batyuka str. 19, 84116 Slavyansk, Ukraine; [email protected] National University of Cherkasy, Physics Department, Laboratory of Mathematical Physics and National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics Generala Batyuka str. 19, 84116 Slavyansk, Ukraine; [email protected] Holon Institute of Technology Golomb St. 52, Holon, 5810201, Israel; [email protected] National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics Generala Batyuka str. 19, 84116 Slavyansk, Ukraine; a.yefi[email protected] In memory of Professor Bogdan Bojarski Abstract. We study the Hilbert boundary value problem for the Beltrami equation in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring–Martio, generally speaking, without (A)-condition by Ladyzhenskaya–Ural’tseva that was standard for boundary value problems in the PDE theory. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of the generalized regular solutions. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundary value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media. 1. Introduction Hilbert [31] studied the boundary value problem formulated as follows: To find an analytic function f(z) in a domain D bounded by a rectifiable Jordan contour C that satisfies the boundary condition (1.1) lim Re {λ(ζ) f(z)} = ϕ(ζ) ∀ ζ ∈ C, z→ζ where both the coefficient λ and the boundary date ϕ of the problem are continuously differentiable with respect to the natural parameter s on C.
    [Show full text]
  • Capacities in Metric Spaces
    Integr. equ. oper. theory 44 (2002) 212-242 0378-620X/02/020212-31 ' I Integral Equations BirkNiuser Verlag, Basel, 2002 and OperatorTheory CAPACITIES IN METRIC SPACES VLADIMIR GOL'DSHTEIN and MARC TROYANOV We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove our results in the ax- iomatic framework of [17]. 1 Introduction Various notions of Sobolev spaces on a metric measure spaces (X, d, #) have been introduced and studied in recent years (see e.g. [4], [18], [23] and [41]). Good sources of information are the books [20] and [21]. The basic idea of all these constructions is to associate to a given function u : X --+ R a collection D[u] of measurable functions g : X --+ N+ which control the variation of u. The Dirichlet p-energy of the function u is then defined as and the Sobolev space is WI'v(X):= {u C Lv(X) I gp(u) < oo}. We say that g is a pseudo-gradient of u if g E D[u] and the correspondence u -+ D[u] (i.e. the way pseudo-gradients are defined) is called a D-structure. The theory of D-structures is axiomatically developped in [17]. In the present paper, we introduce two notions of capacities on a metric measure space X equipped with a D-structure. First the Sobolev capacity of an arbitrary subset F C X is defined to be @(F) := inf{ IlUllwl.p I u E Bp(F) }, where Bp(F) := {u C Wt'P(X)I u _> 1 near F and u > 0 a.e.}.
    [Show full text]